On the Union of Cylinders in Three Dimensions∗
Esther Ezra
Department of Computer Science
Duke University
[email protected]
Abstract
We show that the combinatorial complexity of the union
of n infinite cylinders in R3 , having arbitrary radii, is
O(n2+ε ), for any ε > 0; the bound is almost tight in
the worst case, thus settling a conjecture of Agarwal and
Sharir [3], who established a nearly-quadratic bound for
the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal
and Sharir [3], in particular, a simple specialization of our
analysis to the case of nearly congruent cylinders yields a
nearly-quadratic bound on the complexity of the union in
that case, thus significantly simplifying the analysis in [3].
Finally, we extend our technique to the case of “cigars” of
arbitrary radii (that is, Minkowski sums of line-segments
and balls), and show that the combinatorial complexity of
the union in this case is nearly-quadratic as well. This problem has been studied in [3] for the restricted case where
all cigars are (nearly) equal-radii. Based on our new approach, the proof follows almost verbatim from the analysis
for infinite cylinders, and is significantly simpler than the
proof presented in [3].
1 Introduction
Let K = {K1 , . . . , Kn } be a collection of n infinite
cylinders in R3 . Denote the boundary of Ki by Ci , for
i = 1, . . . , n, and put C = {C1 , . . . , Cn }; with a slight
abuse of notation, we also refer to the elements of C as
cylinders. Let A(C) denote the three-dimensional arrangement induced by the cylinders in C, i.e., the decomposition
of 3-space into vertices, edges, faces, and three-dimensional
cells, each being a maximal connected set contained in the
intersection of a fixed subcollection of the cylinders
of C
n
and not meeting any other cylinder. Let U = i=1 Ki denote the union of K. The combinatorial complexity of U is
∗ Work
on this paper was supported by NSF under grants CNS-0540347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-041-0278 and W911NF-07-1-0376, and by an NIH grant 1P50-GM-0818301.
the number of vertices, edges and faces of the arrangement
A(C) appearing on the union boundary. The problem studied in this paper is to obtain a nearly-quadratic upper bound
on the combinatorial complexity of the union U.
Previous results. The problem of determining the combinatorial complexity of the union of geometric objects has
received considerable attention in the past twenty years, although most of the earlier work has concentrated on the planar case. See [1] for a recent comprehensive survey of the
area.
The case involving pseudodiscs (that is, a collection of
simply connected planar regions, where the boundaries of
any two distinct objects intersect at most twice), arises for
Minkowski sums of a fixed convex object with a set of pairwise disjoint convex objects (which is the problem one faces
in translational motion planning of a convex robot), and has
been studied by Kedem et al. [17]. In this case, the union
has only linear complexity. Matoušek et al. [19, 20] proved
that the union of n α-fat triangles (where each of their angles is at least α) in the plane has only O(n) holes, and
its combinatorial complexity is O(n log log n). The constant of proportionality, which depends on the fatness factor α, has later been improved by Pach and Tardos [23].
Extending the study to the realm of curved objects, Efrat
and Sharir [13] studied the union of planar convex fat objects. Here we say that a planar convex object c is α-fat,
for some fixed α > 1, if there exist two concentric disks,
D ⊆ c ⊆ D , such that the ratio between the radii of D
and D is at most α. In this case, the combinatorial complexity of the union of n such objects, such that the boundaries of each pair of objects intersect in a constant number
of points, is O(n1+ε ), for any ε > 0. See also Efrat and
Katz [11] and Efrat [10] for related (and slightly sharper)
nearly-linear bounds.
In three and higher dimensions, it was shown by
Aronov et al. [6] that the complexity of the union of k convex polyhedra with a total of n facets in R3 is O(k 3 +
nk log k), and it can be Ω(k 3 + nkα(k)) in the worst
case. The bound was improved by Aronov and Sharir [5]
to O(nk log k) (and Ω(nkα(k))) when the given polyhedra
are Minkowski sums of a fixed convex polyhedron with k
pairwise-disjoint convex polyhedra. (This problem arises in
the case of a translating convex polyhedral robot in R3 amid
a collection of polyhedral obstacles.) Boissonnat et al. [7]
proved that the maximum complexity
of the union of n axis
parallel hypercubes in Rd is Θ nd/2 , and that the bound
improves to Θ nd/2 if all hypercubes have the same
size. Pach et al. [22] showed that the combinatorial complexity of the union of n nearly congruent arbitrarily oriented cubes in three dimensions is O(n2+ε ), for any ε > 0
(see also [21] for a subcubic bound on the complexity of
the union of fat wedges in 3-space). Agarwal and Sharir [3]
have shown that the complexity of the union of n congruent
infinite cylinders is O(n2+ε ), for any ε > 0. In fact, the
more general problem studied in [3] involves the union of
the Minkowski sums of n pairwise disjoint triangles with a
ball (where congruent infinite cylinders are obtained when
the triangles become lines), and the nearly quadratic bound
is extended in [3] to this case as well. Aronov et al. [4]
showed that the union complexity of n κ-round objects in
R3 is O(n2+ε ), for any ε > 0, where an object c is κround if for each p ∈ ∂c there exists a ball B ⊂ c that
touches p and its radius is at least κ · diam(c). The bound is
O(n3+ε ), for any ε > 0, for κ-round objects in R4 . Finally, Ezra and Sharir [14] have recently shown that the
complexity of the union of n α-fat tetrahedra (that is, tetrahedra, each of whose four solid angles at its four respective
apices is at least α) of arbitrary sizes in R3 is O(n2+ε ),
for any ε > 0. This result immediately yields a nearlyquadratic bound on the complexity of the union of arbitrary
cubes, and thus generalizes the result of Pach et al. [22],
who showed this bound only for the case where the cubes
have nearly equal size lengths. Each of the above known
nearly-quadratic bounds (for the three-dimensional case) is
almost tight in the worst case.
To recap, all of the above results indicate that the combinatorial complexity of the union in these cases is roughly
“one order of magnitude” smaller than the complexity of the
arrangement that they induce. While considerable progress
has been made on the analysis of unions in three dimensions, the case of the union of infinite cylinders of arbitrary
radii has so far been remained elusive.
Our result. In this paper we make a significant progress
on the problem of bounding the complexity of the union
of infinite cylinders of arbitrary radii in 3-space, and show
a nearly-quadratic bound on this complexity, thus settling
a conjecture of Agarwal and Sharir [3], who showed this
bound only for the case where the cylinders are (nearly)
congruent. Our bound, which is the first known non-trivial
bound for this general problem, is almost tight in the worst
case.
Specifically, we show that the complexity of the union
of n cylinders as above is O(n2+ε ), for any ε > 0, where
the constant of proportionality depends on ε. The analysis
is based on some of the ideas presented in [3, 14], and on
“(1/r)-cuttings”, in order to partition space into triangularprism subcells, so that, on average, the overwhelming majority of the cylinders intersecting a subcell Δ are “good”,
in the sense that they behave as functions within Δ with respect to some direction ρ. Thus vertices of the union that are
incident (only) to good cylinders appear on the boundary of
the “sandwich region” enclosed between the “ρ-lower” envelope of a subset of these functions and the “ρ-upper” envelope of another subset of such functions, and, as shown
in [2, 18], the complexity of this region is nearly-quadratic.
It then only remains to analyze the number of other types
of vertices (incident to some of the few “bad” cylinders that
cross Δ), a task which is handled by the divide-and-conquer
mechanism based on our cutting (see below for details).
The problem studied in this paper is a generalization of
the case where all the cylinders are equal radii - a problem that has been studied by Agarwal and Sharir [3]. We
show that a simple specialization of our analysis to that case
yields the same asymptotic bound on the complexity of the
union as above. The analysis, based on our approach, is
significantly simpler than the analysis in [3], and can thus
replace that of [3]. (Note that we use a variant of some of
the ideas given in [3], however, most of the analysis steps
taken in [3] are no longer needed.)
We extend our analysis to the case of “cigars” of arbitrary radii, that is, Minkowski sums of line-segments and
balls, and show that the bound on the combinatorial complexity of the union is nearly-quadratic in this case as well.
This problem has been studied in [3] for the restricted case
where the cigars are equal-radii. Here too, our analysis is
significantly simpler than that of [3], and, in particular, the
original problem is much easier to extend to this case using
our new approach than the approach of [3].
2 The Complexity of the Union
2.1
Preliminaries and Overview
We first assume that the cylinders in K are in general
position, that is, no axes of two of them are parallel or intersect, no two cylinders are tangent to each other, no curve of
intersection of the boundaries of any two cylinders is tangent to a third one, and no four cylinder boundaries meet.
Following the arguments in [3], this assumption involves
no loss of generality. This general position assumption implies that each vertex of the arrangement of the cylinders
lies on exactly three cylindrical boundaries, and is thus incident upon only a constant number of edges and faces. The
number of edges and faces on the boundary of the union ∂U
that are not incident upon any vertex is O(n2 ). Thus the
combinatorial complexity of U is O(n2 + |V (C)|), where
V (C) is the set of vertices of A(C) that appear on ∂U . Our
main result is:
(a)
(b)
Figure 1. (a) Coins in 3-space arranged in a grid-like form. (b) The decomposition into vertical prism cells.
Theorem 2.1 For any set C of n infinite cylinders in R3 ,
|V (C)| = O(n2+ε ), where the constant of proportionality
depends on ε. The bound is almost tight in the worst case.
It is relatively easy (using standard techniques; see,
e.g., [24]) to construct a set of n infinite cylinders that yield
Ω(n2 ) vertices on the boundary of their union (see also [3]
for further details). We thus devote the remainder of this
section to deriving the upper bound stated in Theorem 2.1.
We note that it is crucial to assume that the cylinders are
infinite. Otherwise, the combinatorial complexity of their
union is Ω(n3 ) in the worst case. Indeed, suppose we have
a set of n cylinders, each of which with a sufficiently large
radius and height that is arbitrarily close to 0. We can now
arrange these cylinders in a (three-dimensional) grid-like
structure, resulting in Ω(n3 ) holes in the union; see Figure 1(a).
2.2
The
problem
overview
decomposition—an
We use a divide-and-conquer approach, based on (1/r)cuttings [9, 8]. Specifically, we project all the cylinders in K
onto the xy-plane, thereby obtaining a set K0 of n infinite
strips. Let L denote the set of their bounding lines. (We
assume that the coordinate system is generic, so none of
these lines projects to a single point.)
We construct a (1/r0 )-cutting of the planar arrangement
A(L), by taking a random sample R0 of O(r0 log r0 ) lines
of L, for some sufficiently large constant parameter r0 , constructing the planar arrangement A(R0 ), and triangulating
each of its cells, using, e.g., bottom-vertex triangulation.
The number of simplices is proportional to the overall complexity of A(R0 ), and is thus O(r0 2 log2 r0 ). The ε-net theory [9, 16] implies that, with high probability, each simplex
of the resulting decomposition is crossed by at most n/r0
lines of L. We pick one sample R0 for which this property
holds, and fix it in the throughout analysis.
We next lift the sub-triangles of each of the cells of
A(R0 ) into vertical prisms. Let Ξ denote the collection of
these prism-cells; see Figure 1(b) for an example.
Our goal is to bound the number of intersection vertices
of the union in each cell of Ξ separately, and then sum these
bounds over all these cells. Fix a cell Δ of Ξ. We classify
each cylinder K ∈ K that intersects Δ as being either wide
in Δ, if the radius r of K satisfies r ≥ w/2, where w is
the width of Δ (that is, the minimum distance between any
pair of parallel (vertical) planes that bound Δ), or narrow,
otherwise. See Figure 2 for an example.
As a consequence, each intersection vertex v of the union
that appears in Δ is classified as either WWW, if all three
cylinders that are incident to v are wide in Δ, WWN, if two
of them are wide and one is narrow, WNN, if one of them
is wide and two are narrow, or NNN, if all of them are narrow in Δ. In all four cases, the three relevant cylinders are
distinct.
Let Δ be a prism-cell of Ξ. We first observe that if the
radius r of a cylinder K that meets Δ is smaller than w/2
(that is, K is narrow in Δ), at least one of the two vertical
tangency generator lines (also known as the silhouette) of
K must intersect Δ. Indeed, if both lines do not meet Δ,
the diameter d := 2r of K must be larger than w, as is
easily verified, contrary to the assumption that K is narrow
in Δ. We thus charge the crossing of K and Δ to that of
and Δ, and since L is in fact the set of the projections of
the silhouette lines onto the xy-plane, we conclude that Δ
meets at most rn0 narrow cylinders; see once again Figure 2.
In what follows, we denote by Δ a fixed prism-cell of
Ξ, and by W = W Δ (resp., N = N Δ ) the set of the wide
Δ
:=
(resp., narrow) cylinders within Δ. Put MW = MW
Δ
Δ
Δ
|W |, and MN = MN := |N |. As just discussed,
MW ≤ n, MN ≤ rn0 .
2.3
The overall recursive analysis
We now present a recursive decomposition, where in
each step, we are given a (parent) cell Δ0 and a set
N Δ0 of narrow cylinders that meet Δ0 , and we construct
O(r0 2 log2 r0 ) prism-subcells Δ of Δ0 as above. At the first
2
K
1
2r
0
2
Δ0
Δ
0
1
(a)
(b)
Figure 2. The cylinder K is wide within the prism-cell Δ. (a) The pair of the silhouette lines 1 , 2 of K do not meet
Δ. (b) The projection of this scene onto the xy-plane. The lines 01 , 02 are the respective projections of 1 , 2 , Δ0 is the
projection of Δ, and r is the radius of K. The strip bounded by 01 , 02 contains Δ0 in its interior.
recursive step, Δ0 equals to the entire three-dimensional
space, and N Δ0 is the entire set of the input cylinders (and
thus |N Δ0 | = n at that step). As the space is progressively
cut up into subcells, more and more narrow cylinders become wide, and the size of the set N Δ0 keeps decreasing.
During each step of the recursion, we immediately dispose
of any new WWW-, WWN-, and WNN-vertices within each
subcell Δ (that is, vertices that were NNN in the parent cell
Δ0 ), and continue to bound the number of the remaining
NNN-vertices recursively. In particular, this implies that we
dispose of all the new wide cylinders (that is, cylinders that
were narrow in Δ0 ) in a single step, and thus we do not keep
processing them in any further recursive step, but process
only the (remaining) narrow ones. The recursion bottoms
out when MN ≤ c, for some absolute constant c ≥ 3. In
this case the number of the remaining intersection vertices
of the union within the current cell Δ0 is O(1).
We show below that the overall number
of
the
Δ of vertices
Δ 2+ε
, for
+ MW
)
first three types in a subcell Δ is O (MN
any ε > 0. This implies that the overall number of new
WWW-, WWN-, and WNN-vertices
generated within each
Δ0 2+ε
Δ0
Δ
subcell Δ of Δ0 is O (MN
)
≤ MN
,
(since MW
Δ0
M
Δ0 2+ε
Δ
=
≤ rN0 ), for a total of O r0 2 log2 r0 (MN
)
MN
Δ0 2+ε
O (MN )
such vertices (recall that r0 is constant),
over all subcells of Δ0 . Let U0 (MN ) denote the maximum
number of intersection vertices that appear on the boundary
of the union at a recursive step involving up to MN narrow
cylinders. Then U0 satisfies the following recurrence:
U0 (MN ) ≤
8 `
´
2+ε
>
O MN
+ O(r0 2 log2 r0 )U0 (MN /r0 ),
>
>
>
>
>
if MN > c,
<
>
>
>
O(1),
>
>
>
:
if MN ≤ c,
where ε > 0 is arbitrary, c ≥ 3 is an appropriate constant
as above, and the constant of proportionality in the nonrecursive term depends on r0 (and on ε).
Using induction on MN , it is easy to verify that the so2+ε
lution of this recurrence is U0 (MN ) = O(MN
), for any
ε > 0, slightly larger than the ε in the non-recursive term,
but still arbitrarily close to 0, with a constant of proportionality that depends on ε. Substituting the initial value
MN = n, we conclude that the overall number of vertices
of the union is O(n2+ε ), for any ε > 0, as asserted.
We thus devote the remainder of this section to deriving
the bound, stated above, on the number of vertices of the
first three types, which is the major part of the analysis.
The number of WWW-vertices of the union. Our next
goal is to show that the number of WWW-vertices of the
union, contained in the fixed prism-cell Δ, is only nearlyquadratic.
Let H, H be the pair of (parallel and vertical) planes
that determine the width w of Δ. We use a simple variant
of the analysis given in [3], in order to show that the WWWvertices of Δ (in fact, it is sufficient to assume that they lie
in the slab bounded by H, H ) appear on the boundary of
the sandwich region enclosed between a lower envelope of
a collection of portions of the wide cylinders and an upper
envelope of another such collection. The analysis of [2]
(see also [18]) implies that this complexity is only nearlyquadratic. For the sake of completeness, we repeat some of
the details given in [3], and modify them to fit our setup.
Let κ be a sufficiently large constant, whose value will
be fixed shortly. We partition each of the wide cylinder
boundaries into κ canonical strips (bounded by generator
lines, parallel to the cylinder axis), each with an angular
span of 2π/κ in the cylindrical coordinate frame induced
by the cylinder.
Let S2 be the unit sphere of directions in R3 . We say that
a direction ρ ∈ S2 is good for a strip τ if the following two
conditions hold.
(i) The angle between ρ and the (outer) normal n of either
of the planes H, H is smaller than π/2 − π/κ.
(ii) Each line tangent to (the relative interior of) τ forms an
angle of at least π/κ with ρ.
In other words, the first condition implies that the inner
product ρ · n is sufficiently large (in particular, they are not
orthogonal), and the second condition implies that when we
enter into the cylinder K, which contains τ as a bounding
strip, in the ρ-direction from a point on τ , we do not remain close to the boundary of K (and, in particular, to τ ).
Intuitively, these two conditions imply that when we move
in the ρ-direction, we expect to leave the strip enclosed between H, H relatively soon, but make a sufficiently long
way inside K.
We say that ρ is a good direction for a vertex v of the
union, incident upon three canonical strips, if it is good for
each of these strips; see Figure 3. As observed in [3], the set
Bτ of bad directions for a fixed strip τ is the union B1 ∪B2 ,
where
(a) B1 is contained in a spherical band consisting of all
points lying at spherical distance at most π/κ from the great
circle on S2 normal to n (in fact, this circle is obtained by
intersecting S2 with a plane parallel to H (or H ) through
its center). The area of B1 is 4π sin (π/κ).
(b) B2 is contained in a spherical band consisting of all
points lying at spherical distance at most 2π/κ from a great
circle on S2 ; see [3] for the easy proof. The area of B2 is
thus 4π sin (2π/κ).
It thus follows (see [3] for further details) that the
area of the set of good directions for v is at least
4π [1 − 3 sin (2π/κ) − sin (π/κ)].
As observed in [3], the set of the good directions contains a spherical cap of some constant opening angle δ, if κ
is sufficiently large. It then implies that each vertex v on the
boundary of the union
at least one good direction, taken
has
from a set Z of O 1/δ 2 points on S2 , for some sufficiently
small value of δ that depends on κ. Note that the above considerations do not assume that v is a vertex of type WWW.
For each fixed ρ ∈ Z, let VΔ (ρ) be the subset of all vertices
(of any of the above four types) of the union that lie inside
Δ, for which ρ is a good direction.
We now partition the slab σ (and thus Δ) enclosed be1
equal subslabs σ by adding
tween H, H into t > sin2 (π/κ)
t planes parallel to H inside σ at equal distances. Our next
goal is to show that the strips τ , for which ρ is a good direction, behave as functions (with respect to that direction)
within σ . The following lemma is a simple variant of [3,
Lemma 2.9].
Lemma 2.2 Let σ be a subslab of σ, and let v be a WWWvertex of the union, contained in σ ∩ Δ and incident upon
three strips τ1 , τ2 , τ3 . Let ρ ∈ Z be a good direction for v.
Then any line parallel to ρ intersects τ1 in at most one point.
Moreover, for any point u ∈ τ1 ∩ σ , in which we enter into
the cylinder K1 bounded by τ1 in the direction parallel to
ρ, we reach ∂σ before exiting K1 . Similar properties hold
for τ2 , τ3 .
Proof: Suppose first that there is a line parallel to ρ that
intersects τ1 twice. This would imply that τ1 must have a
tangent in the ρ-direction, as is easily verified, contrary to
the assumption that ρ is a good direction.
As to the second assertion of the lemma, let u be the
other intersection point of ∂K1 and the line passing through
u and parallel to ρ. It is easy to verify that |uu | is minimized when uu is orthogonal to the axis of K1 and forms
an angle π/κ with the tangent plane to K1 at u. It thus follows that |uu | ≥ 2r1 sin (π/κ), where r1 is the radius of
K1 . On the other hand, since uu forms an angle of at most
π/2 − π/κ with the normal n (to either of the two bounding planes of σ ), it follows that the length of the projection
of uu on n is at least |uu | sin (π/κ) ≥ 2r1 sin2 (π/κ) ≥
w sin2 (π/κ), where w is the width of σ, as above. Thus
1
, u must lie outside σ . This
if we choose t > sin2 (π/κ)
completes the proof of the lemma. 2
We now denote by Tσ (ρ), the set of canonical strips
τ that cross the subslab σ and have at least one vertex
in VΔ (ρ). We clip all the strips to Δ ∩ σ . A (clipped)
strip τ in Tσ (ρ) is ρ-upper (resp., ρ-lower) if, for any point
u ∈ τ , the point u+αρ lies in the exterior (resp., interior) of
the cylinder K whose boundary contains τ , for sufficiently
small values of α > 0. Let Tσ+ (ρ) (resp., Tσ− (ρ)) be the
set of the ρ-upper (resp., ρ-lower) strips of Tσ (ρ). The ρupper envelope of Tσ+ (ρ) is the set of points u on the strips
of Tσ+ (ρ), such that a ray from u in the (+ρ)-direction does
not meet any other strip in Tσ+ (ρ). The ρ-lower envelope of
Tσ− (ρ) is defined analogously.
Following the analysis of [3] and Lemma 2.2, each
WWW-vertex v ∈ VΔ (ρ) must lie on the boundary of the
sandwich region enclosed between the ρ-upper envelope of
the strips in Tσ+ (ρ) and the ρ-lower envelope of the strips
in Tσ− (ρ), and, according tothe results
of [2, 18], the num2+ε
, for any ε > 0, with a
ber of these vertices v is O MW
constant of proportionality that depends on κ and ε. Repeating the analysis for all subslabs σ , and all good directions
ρ ∈ Z, we obtain a similar bound on the overall number of
WWW-vertices of the union that are contained in Δ.
Remark: We note that it is crucial to process only the strips
that have at least one vertex v of the union for which ρ is a
good direction, and construct the ρ-lower and ρ-upper envelopes only with respect to these strips. All the remaining
strips should not be considered at that step, as they may
violate the envelope properties. In particular, we ignore
wide cylinders whose corresponding strips do not contain
any such vertices v.
τ1
τ2
ρ
n
v
τ3
H
H
Figure 3. The vector ρ is a good direction for the vertex v of the union, incident upon three cylinder strips τ1 , τ2 , τ3 ,
where n is the normal to the planes H, H .
The number of WWN-vertices of the union. We next
establish a nearly-quadratic bound on the number of WWNvertices of the union, with respect to a fixed prism-cell Δ.
Let us first fix a good direction ρ ∈ Z and a subslab σ . We
bound the number of WWN-vertices in each subcell Δ =
Δ ∩ σ separately, and then sum these bounds over all these
subcells. Note that some of the narrow cylinders within Δ
may become wide in Δ , nevertheless, we regard them as
narrow; this does not effect the analysis. We thus have, at
each step of the analysis, a prism-cell Δ (confined to a slab
σ ), a set W = W Δ of MW wide cylinders, and a set N =
Δ
N of MN narrow cylinders.
Since each WWN-vertex v ∈ VΔ (ρ) (where VΔ (ρ) is
the subset of all vertices of the union that lie inside Δ for
which ρ is a good direction) involves a wide cylinder, it
must lie on the boundary of the sandwich region enclosed
between the ρ-upper and ρ-lower envelopes of the strips in
Tσ+ (ρ) and Tσ− (ρ), respectively. Moreover, v is obtained
as the intersection of an edge of the sandwich region and a
narrow cylinder in Δ .
We now apply a second decomposition step, in which
we partition Δ into subprisms Δ (clipped to within Δ ),
so that each of them contains a relatively small number of
both narrow and wide cylinders, as follows.
We draw two random samples R+ (ρ) ⊆ Tσ+ (ρ),
−
R (ρ) ⊆ Tσ− (ρ) of O(r log r) strips each, for some sufficiently large constant parameter r, and construct the overlay
Mρ of the respective minimization diagrams of the ρ-upper
and ρ-lower envelopes of the strips in R+ (ρ) and R− (ρ);
these diagrams are obtained on a plane Hρ⊥ perpendicular
to the ρ-direction. By the results of [2, 18], the overall complexity of Mρ is O(r2+ε ), for any ε > 0. We next refine the
overlay Mρ by projecting the narrow cylinders in Δ onto
the plane Hρ⊥ , taking a random sample R of O(r log r)
of their bounding lines1 , and then overlaying Mρ with the
lines in R; we thus draw three random samples in total. Let
MR
ρ be the refined decomposition. We first show:
Proof: The overall complexity
of the arrangement A(R) of
the lines in R is O r2 log2 r , and, as noted above, the overall complexity of Mρ is O(r2+ε ), for any ε > 0. We next
show that the overall number of edge-crossings between
Mρ and A(R) is O(r log rλ6 (r log r)) = O(r2 polylogr),
where λs (q) is the maximal length of Davenport-Schinzel
sequences of order s on q symbols (see [24]). This will assert the bound in the lemma.
Indeed, let HR be the set of the O(r log r) planes containing the lines in R and parallel to the ρ-direction. Each
plane H ∈ HR intersects each of the strips in R+ (ρ),
R− (ρ) in an elliptic arc (clipped to within Δ ). Let Γ+
H
be the set of these ρ-upper arcs, and let Γ−
H be the set of
these ρ-lower arcs. Since H is parallel to the ρ-direction,
it must meet each of the edges e on the boundary of the
sandwich region, enclosed between the ρ-upper envelope of
the strips in R+ (ρ) and the ρ-lower envelope of the strips
in R− (ρ), in a vertex that appears along the boundary of
the sandwich region enclosed between the upper envelope
−
of the arcs in Γ+
H and the lower envelope of the arcs in ΓH
(in the coordinate frame of H). The overall number of these
vertices is O(λ6 (r log r)), since each pair of such arcs intersect in at most four points (see, e.g., [24] for details).
Each of these vertices is projected on the plane Hρ⊥ , perpendicular to the ρ-direction, to an edge-crossing between
Mρ and A(R). In fact, by similar arguments, we can conclude that the overall number of edges e of either of the ρupper envelope of the strips in R+ (ρ), or ρ-lower envelope
of the strips in R− (ρ) that meet H is also O(λ6 (r log r))
1 From technical reasons, at the first step of this decomposition we need
to ignore the cylinders that were narrow in Δ and became wide in Δ .
This does not affect the analysis, as these cylinders are wide at the next
decomposition step.
Lemma 2.3 The overall number of vertices in MR
ρ is
O(r2+ε ), for any ε > 0.
(for technical reasons, we need that latter bound, since Mρ
corresponds to the overlay of these two envelopes, which
may somewhat be different from the sandwich region enclosed between these two envelopes). Summing this bound
over all planes H ∈ HR yields the asserted bound. 2
We next triangulate each of the cells of MR
ρ (in the coordinate frame of Hρ⊥ ), and lift each of these simplices in the
ρ-direction, thereby obtaining a collection Ξ of O(r2+ε ) triangular prism-subcells. We collect all these prisms that lie
in the sandwich region enclosed between the ρ-upper envelope of the strips in R+ (ρ) and the ρ-lower envelope of
the strips in R− (ρ). Each of these prisms is bounded in the
ρ− - and ρ+ -directions (either by the boundary of Δ or) by
strips τ + ∈ R+ (ρ) appearing on the ρ-upper envelope, and
τ − ∈ R− (ρ) appearing on the ρ-lower envelope, respectively. We thus obtain a decomposition of the above sandwich region (within Δ ) into O(r2+ε ) triangular prisms.
We observe that each cylinder K that is wide in Δ continues to be wide in each clipped prism-subcell Δ , since
the width of Δ cannot exceed that of Δ (or of Δ). By the
cutting properties, with high probability, each prism-subcell
Δ of the resulting decomposition is crossed by at most
O(MW /r) strips in Tσ+ (ρ) and by at most O(MW /r) strips
in Tσ− (ρ), with a constant of proportionality that depends
linearly on κ. We pick a pair of samples R+ (ρ), R− (ρ) for
which this property holds, and fix them in the throughout
analysis.
We next bound the number of wide cylinders within a
prism-subcell Δ . The strips in Tσ+ (ρ) ∪ Tσ− (ρ) are said to
be good, and all the remaining strips (of the wide cylinders
within Δ ) are said to be bad. Clearly, the number of wide
cylinders K in Δ that have at least one good strip that
meets Δ is at most MW /r, since we can charge K to that
good strip. If K has only bad strips that meet Δ , we ignore
K in the subsequent analysis of Δ . This does not violate
the bound on the number of (original) WWN-vertices v in
Δ , since each such vertex is obtained on the boundary of
the sandwich region enclosed between the ρ-upper and ρlower envelopes of the good strips, and thus v does not lie
on ∂K ∩ Δ . We thus continue the subsequent analysis
within Δ with at most MW /r wide cylinders.
Each cylinder K that is narrow in Δ either becomes
wide in Δ , or continues to be narrow in that cell. Let
Lρ be the set of the ρ-silhouette lines of the narrow cylinders within Δ (that is, generator lines in the ρ-direction).
Applying similar arguments as above, a narrow cylinder K
within a prism-subcell Δ of the decomposition must have
a ρ-silhouette line , whose projection on Hρ⊥ meets the
projection of Δ on that plane. We thus charge the crossing of K and Δ to that of the respective projections of and Δ , and conclude that the number of narrow cylinders
within Δ is at most MN /r, with high probability. We pick
one sample R for which this property holds, and fix it in the
throughout analysis. We have thus shown:
Lemma 2.4 The number of wide cylinders within a subcell
Δ of Δ that contribute WWN-vertices to the union is at
most MW /r, and the number of narrow cylinders in that
subcell is at most MN /r.
We bound the number of WWN-vertices of the union by
applying a recursive decomposition scheme, where in each
step we construct O(r2+ε ) prism-cells Δ , using (1/r)cutting of the good strips (taken from the current set of
wide cylinders that we process in Δ ) and the boundary
projections of the narrow cylinders, as above, where each
subproblem consists of at most MW /r wide cylinders, and
at most MN /r narrow cylinders. In each step we dispose immediately of all the new WWW-vertices in Δ (that
is, vertices that were WWN in the parent cell Δ , and
have just became WWW) and continue to process in recursion the remaining WWN-vertices. Thus cylinders that
were narrow in the parent cell of Δ and became wide in
Δ need not be processed in any further recursive step,
the only wide cylinders that we process are those of the
original problem. The recursion bottoms out when either
MW ≤ c, or MN ≤ c, for some absolute constant c ≥ 3.
We then bound by brute-force the number of the (remaining) WWN-vertices, and thus obtain an overall bound of
2
2
MN ) = O(MW
+ MN ) on the number of these
O(MW
vertices.
WWW-vertices in Δ is
The number of1+δnew
O (MN + MW )MW , for any δ > 0. Indeed, the
number of new wide cylinders within Δ is at most MN ,
and the number of the old wide cylinders (cylinders that
were wide in the parent cell of Δ ) is at most MW . Assume
first that MN ≤ MW . Following the considerations given
in the WWW-case, we conclude that the number of new
2+δ
WWW-vertices in Δ is O(MW
), for any δ > 0. If
MN > MW , we partition the set of the narrow cylinders
MN
(roughly) equal subsets, each of which coninto M
W
taining at most MW elements. Each new WWW-vertex
v involves a pair of old wide cylinders and a new wide
cylinder within Δ . We thus apply the nearly-quadratic
bound in Δ on the set of the old wide cylinders and
each of the subsets of new wide cylinders separately, and
then sum these bounds over all subsets, thereby obtaining
1+δ
), for any δ > 0 (see
an overall bound of O(MN MW
also [3, 14] for similar considerations). The bound now
follows. Since r is constant, we obtain that the overall
number of new
over
all prism-subcells
WWW-vertices,
1+δ
.
Δ , is also O (MN + MW )MW
Let U1 (MN , MW ) denote the maximum number of
WWN-vertices that appear on the boundary of the union
at a recursive step involving up to MN narrow cylinders,
and MW wide cylinders. Then U1 satisfies the following
recurrence:
8 `
´
1+δ
>
O (MN + MW
)MW
>
“
”+
>
>
MN MW
>
2+ε
>
,
O(r
)U
,
1
>
r
r
>
>
<
if min{MN , MW } > c,
U1 (MN , MW ) ≤
>
>
>
>
>
2
>
),
>O(MN + MW
>
>
:
if min{MN , MW } ≤ c,
where ε > 0, δ > 0 are arbitrary, c ≥ 3 is an appropriate
constant, and where the constant of proportionality in the
first expression depends on (ε, δ, κ and on) r. It is straightforward to verify (see also [15], [14]), that the solution of
1+ε
),
this recurrence is U1 (MN , MW ) = O((MN +MW )MW
for any ε > 0 (slightly larger than δ in the non-recursive
term and the ε in the recursive term, but still arbitrarily close
to 0), with a constant of proportionality that depends on ε
and on κ.
Summing this bound over all Δ = Δ ∩ σ , and over all
good directions ρ ∈ Z, we obtain that the overall number
1+ε
, for any
of WWN-vertices in Δ is O (MN + MW )MW
ε > 0, with a constant of proportionality that depends on ε
and κ.
Remark: The recursive mechanism is such that we sample
the strips that belong to the wide cylinders and not the actual
wide cylinders, since we have to take only those strips that
have vertices v of the union, for which ρ is a good direction,
and then recursively construct out of them the sandwich region that they induce. The role of the wide cylinders is in
bounding the number of new WWW-vertices created in a
subcell Δ of the current cell Δ . We thus recurse with
the wide cylinders, but construct the cutting with their good
strips.
The number of WNN-vertices of the union. As in the
WWN-case, we fix a good direction ρ ∈ Z and a subslab
σ , and bound the number of WNN-vertices in each subcell Δ = Δ ∩ σ separately. We then sum these bounds
over all these subcells, and over all ρ. Here too, we observe
that since each WNN-vertex v ∈ VΔ (ρ) involves a wide
cylinder, it must lie on the boundary of the sandwich region
enclosed between the ρ-upper and ρ-lower envelopes of the
strips in Tσ+ (ρ) and Tσ− (ρ), respectively.
Using similar notations as in the WWN-case, we apply
a similar decomposition, where we recursively construct
O(r2+ε ) cells Δ , each of which meets at most MW /r
wide cylinders and at most MN /r narrow cylinders. At
each recursive step, we dispose immediately of all the new
WWW- and WWN-vertices (that is, vertices v that were
WNN-vertices at the parent cell of Δ , and have become either WWW- or WWN-vertices at Δ ), and continue to process in recursion the (remaining) WNN-vertices. Applying
similar considerations as above, the overall number of new
WWW in the subcells Δ , over all Δ ,
and WWN-vertices
1+δ
, for any δ > 0. At the bottom of
is O (MN + MW )MN
the recursion (when either MW ≤ c, or MN ≤ c, for some
absolute constant c ≥ 3), we bound by brute-force the number of the (remaining) WNN-vertices, obtaining an overall
2
2
bound of O(MW MN
) = O(MW + MN
) on the number
of these vertices. Letting U2 (MN , MW ) denote the maximum number of WNN-vertices that appear on the boundary
of the union at a recursive step involving up to MN narrow
cylinders and MW wide cylinders, we have:
8 `
´
1+δ
>
O (MN + MW
)MN
>
“
”+
>
>
MN MW
>
2+ε
>
,
)U
,
O(r
2
>
r
r
>
>
<
if min{MN , MW } > c,
U2 (MN , MW ) ≤
>
>
>
>
>
2
>
+ MW ),
>O(MN
>
>
:
if min{MN , MW } ≤ c,
where, as above, ε > 0, δ > 0 are arbitrary, c ≥ 3 is
an appropriate constant, and where the constant of proportionality in the first expression depends on (ε, δ, κ and on)
r. Here too, it is easy to verify that
this
re the solution of1+ε
, for
currence is U2 (MN , MW ) = O (MN + MW )MN
any ε > 0 (slightly larger than δ in the non-recursive term
and the ε in the recursive term, but still arbitrarily close to
0), with a constant of proportionality that depends on ε and
on κ.
Remarks: (1) The WWN- and WNN-cases, albeit being
similar, cannot be handled at the same recursive step, and
need to be analyzed separately. When a WNN-vertex v
becomes WWN in a subcell Δ , we need to consider the
envelopes with respect to a (possibly, new) good direction
ρ , not necessarily equal to ρ. These envelopes correspond
to the old and the new wide cylinders in Δ , and we need
to continue with the recursive decomposition described for
the WWN-case, according to the direction ρ , with the set of
these wide cylinders and the narrow ones that have survived
when passing from Δ to Δ .
(2) The only part of the analysis in which we explicitly use
the geometric structure of the cylinders is in the WWWcase. The analysis for the WWN- WNN-cases only uses the
property that the vertices lie on the boundary of the sandwich region enclosed between two envelopes of the good
strips.
The case of congruent cylinders. We now study the case
where all the cylinders have equal radii, and show that a
simple specialization of our approach leads to a nearlyquadratic bound on the complexity of their union.
We only need to make the following simple observation.
We apply the decomposition described in Section 2.2, and
observe that, since all cylinders are equal radii, all of them
are either wide or narrow in a fixed cell Δ of the decomposition. Thus all vertices of the union inside Δ are either of
type WWW or of type NNN. We thus construct this decomposition recursively, where in each step we dispose of all the
new WWW-vertices within each subcell Δ, and continue to
bound the number of NNN-vertices recursively. Here we do
not need to apply the second decomposition step described
for the WWN- and WNN-cases.
We note that these arguments can easily be extended to
the case of nearly-congruent cylinders, that is, when the
radii of the cylinders are different, but vary between some
constant α < 1 and 1. We omit the easy proof in this version.
3 An extension to the case of cigars.
We now extend the analysis to the following case. We
are given a set S = {s1 , . . . , sn } of n line-segments in 3space and a set B = {B1 , . . . , Bn } of n balls of arbitrary
radii. The set C = {s1 ⊕ B1 , . . . , sn ⊕ Bn } consisting
of the Minkowski sums of si ∈ S and Bi ∈ B (that is,
si ⊕ Bi = {x + y | x ∈ si , y ∈ Bi }), for i = 1, . . . , n,
is the input set to our problem. The elements of C are also
referred to as cigars in this case. We show:
Theorem 3.1 The combinatorial complexity of the union of
n cigars of arbitrary radii in 3-space is O(n2+ε ), for any
ε > 0, where the constant of proportionality depends on ε.
The bound is almost tight in the worst case.
We devote this section to the proof of Theorem 3.1.
We first assume that the cigars in C are in general position. As discussed in [3], this excludes the cases where no
pair of the segments in S are parallel or intersect, that no
two cigars in C are tangent to each other, that no curve of
intersection of the boundaries of any two cigars is tangent
to a third one, that no triple intersection of the boundaries
of the cigars lie on any circle separating the cylindrical and
spherical portions of one of them, and that no four boundaries meet at a point.
As a result, each vertex v of the union U = C∈C C is
the intersection point of either (i) three cylindrical boundaries, (ii) a pair of cylindrical boundaries and one spherical
boundary, (iii) a cylindrical boundary and a pair of spherical
boundaries, or (iv) three spherical boundaries. We refer to
each of these vertices as being of type CCC, CCS, CSS, or
SSS, respectively. In the analysis of [3], these four cases are
handled separately, however, using our new approach, we
show that all these cases can be analyzed simultaneously.
Note that in the SSS-case, each vertex v of the union lies on
the boundary of the union of three distinct spheres, and thus
the problem is reduced in this case to bounding the complexity of the union of 2n balls in 3-space, which is known
to be O(n2 ) (see, e.g., [24]), however, this case is subsumed
by our analysis.
We now apply a similar decomposition scheme to that
of Section 2.2. That is, we project all the cigars in C onto
the xy-plane, and construct a (1/r0 )-cutting for the boundaries of these projections, for some sufficiently large con-
stant parameter r0 , thereby obtaining a set of O(r0 2 log2 r0 )
pseudo-triangles, which we lift into vertical prisms.
We next use a similar definition of wideness as in the
case of cylinders. That is, a cigar C ∈ C that intersects a
prism-cell Δ is wide in Δ, if its radius r is at least w/2
(where w is the width of Δ), and narrow, otherwise. Here
too, it is easy to verify that the silhouette of a narrow cigar
in a cell Δ must cross that cell.
We apply a similar recursive mechanism as in Section 2.3. The major difference in the analysis with respect to
the case of cylinders is in bounding the number of WWWvertices. We use here similar ideas to those presented in [3]
for bounding the complexity of cigars of equal radii. Let κ
be the same constant as in Section 2.3. We partition each of
the cylindrical boundaries of the cigars C into κ canonical
strips as before, and we cover (both) hemispherical portions
of C by O(κ2 ) spherical caps, each of opening angle at most
π/κ, so that no point lies in more than a constant number of
caps. We define a good direction for a cap in a similar manner as for strips (see conditions (i)–(ii) in the WWW-case in
Section 2.3). The set of bad directions for such a spherical
cap τ is again the union B1 ∪ B2 , where B1 is defined as
in the case of cylinders, and B2 is defined as the spherical
band consisting of all points at spherical distance at most
2π/κ from a great circle Bτ on S2 , where Bτ corresponds
to the plane parallel to the tangent plane of the cap τ at its
center.
Following the reasonings in Section 2.3, each vertex v of
the union has at least one good direction, taken from a set Z
of O(1) directions. It is now easy to verify that the analog to
Lemma 2.2 in this extended case continues to hold as well.
Let us fix a prism-cell Δ of the decomposition and a
good direction ρ, and let σ, σ , VΔ (ρ), Tσ+ (ρ), Tσ− (ρ) be
as in Section 2.3. Following these notations, let Sσ+ (ρ) be
the set of the ρ-upper spherical caps that cross the subslab
σ and contribute at least one vertex to VΔ (ρ). We define
Sσ− (ρ) in an analogous manner. It now follows that each
vertex v ∈ VΔ (ρ) must appear on the boundary of the sandwich region enclosed between the ρ-upper envelope of the
strips and caps in Tσ+ (ρ) ∪ Sσ+ (ρ) and the ρ-lower envelope of the strips and caps in Tσ− (ρ) ∪ Sσ− (ρ). Repeating
the analysis for all good directions ρ, we obtain a nearlyquadratic bound on the overall number of WWW-vertices
of the union in a prism-cell Δ.
The remaining steps of the analysis follow almost verbatim from the analysis for the case of cylinders. This concludes the proof of Theorem 3.1.
4 Concluding Remarks and Open Problems
The only part of the analysis in which we have explicitly used the geometric structure of the cylinders is in the
WWW-case. The analysis in the WWN- WNN-cases only
uses the property that the vertices lie on the boundary of
the sandwich region enclosed between two envelopes of the
good strips, where the major step of the analysis in these
cases relies on the decomposition that we apply. Thus our
machinery can be extended to any set of bodies in 3-space,
for which there is a natural definition for wideness (reps.,
narrowness), from which one can derive similar properties
to those of Lemma 2.2. One such problem concerns the
union of cones. Another related problem is to extend our
machinery from cigars to kreplach, where in this setting, we
modify the definitions from Section 3, so that now S consists of n pairwise-disjoint triangles. This problem was addressed by Agarwal and Sharir [3], who presented a nearlyquadratic bound on the complexity of the union where the
balls in the Minkowski sum have the same radii. An open
problem is extend this bound for the case where the balls
have arbitrary radii.
Acknowledgments. The author wishes to thank Micha
Sharir for helpful discussions and for his useful comments,
and, in particular, for his help in writing this paper. The
author also wishes to thank Boris Aronov and Pankaj Agarwal for useful discussions concerning the problem, and, in
particular, for Boris’ useful comments about this paper,
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